We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair $(G,K)$, where $G={\mathrm{Sp}}_4(\mathbb{R})$ and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2.

Further, by explicating the algebraic structure of the relevant space of $\mathfrak{n}$-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight ${det}^{\ell }{\mathrm{sym}}^m$ with respect to an arbitrary congruence subgroup of ${\mathrm{Sp}}_4(\mathbb{Q})$. We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight ${det}^{{\ell }^{\prime }}{\mathrm{sym}}^{m^{\prime }}$ with $({\ell }^{\prime },m^{\prime })$ varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight ${det}^3{\mathrm{sym}}^{m^{\prime }}$ that cannot be obtained from holomorphic forms.

As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction.

On a smooth projective threefold, we construct an essentially surjective functor $\mathcal{F}$ from a category of two-term complexes to a category of quotients of coherent sheaves and describe the fibers of this functor. Under a coprime assumption on rank and degree, the domain of $\mathcal{F}$ coincides with the category of higher-rank PT-stable objects, which appears on one side of Toda’s higher-rank DT/PT correspondence formula. The codomain of $\mathcal{F}$ is the category of objects that appears on one side of a correspondence formula by Gholampour and Kool, between the generating series of topological Euler characteristics of two types of quot schemes.

As a natural extension of the theory of uniform vector bundles on Fano manifolds, we consider uniform principal bundles, and study them by means of the associated flag bundles, as their natural projective geometric realizations. In this paper, we develop the necessary background and prove some theorems that are flag bundle counterparts of some of the central results in the theory of uniform vector bundles.

Given a group acting on a Gromov hyperbolic space, Bestvina and Fujiwara introduced the WPD property—weak proper discontinuity—for studying the second bounded cohomology of the group. We carry out a more general study of second bounded cohomology using a really weak proper discontinuity property known as WWPD that was introduced by Bestvina, Bromberg, and Fujiwara.

We propose an extended notion of adelic $\mathbb{R}$-Cartier divisors, called ${\ell }^1$-adelic $\mathbb{R}$-Cartier divisors, which enables us to associate a Banach space to each algebraic variety over the field of rational numbers, and establish the global continuity of the arithmetic volume function defined on the space of pairs of ${\ell }^1$-adelic $\mathbb{R}$-Cartier divisors and $\mathbb{R}$-base conditions.